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**A Torelli-type theorem for gravitational instantons.**
*(English)*
Zbl 0671.53046

In ibid. 29, 665-683 (1989; see above) the author gave a natural construction for asymptotically locally Euclidean hyper-Kähler 4- manifolds. In this paper he proves that his construction gives all such manifolds.

To do this, he makes use of the Penrose twistor space Z of such a manifold. This is a 3-dimensional complex manifold whose holomorphic geometry determines the hyper-Kähler metric. The author’s proof first of all makes use of the ALE property to compactify the twistor space as an orbifold by adding a projective line at infinity. He then studies the structure of the graded ring \(A(Z)=H^ 0(Z;{\mathcal O}(k))\) making use of vanishing theorems for higher cohomology groups which are implied by the Penrose transform which relates solutions of the zero rest mass field equations on the ALE space to sheaf cohomology groups on Z. The algebraic structure of A(Z) and its relation to the line at infinity then gives algebraic equations for a singular model of Z which effectively determine the twistor space and its relationship to resolutions of quotient singularities. The “Torelli theorem” in the title refers to the fact that the hyper-Kähler metric on the ALE space is determined up to isometry by the periods of the covariant constant 2-forms \(\omega_ 1\), \(\omega_ 2\) and \(\omega_ 3\), a corollary of the completeness of the result.

To do this, he makes use of the Penrose twistor space Z of such a manifold. This is a 3-dimensional complex manifold whose holomorphic geometry determines the hyper-Kähler metric. The author’s proof first of all makes use of the ALE property to compactify the twistor space as an orbifold by adding a projective line at infinity. He then studies the structure of the graded ring \(A(Z)=H^ 0(Z;{\mathcal O}(k))\) making use of vanishing theorems for higher cohomology groups which are implied by the Penrose transform which relates solutions of the zero rest mass field equations on the ALE space to sheaf cohomology groups on Z. The algebraic structure of A(Z) and its relation to the line at infinity then gives algebraic equations for a singular model of Z which effectively determine the twistor space and its relationship to resolutions of quotient singularities. The “Torelli theorem” in the title refers to the fact that the hyper-Kähler metric on the ALE space is determined up to isometry by the periods of the covariant constant 2-forms \(\omega_ 1\), \(\omega_ 2\) and \(\omega_ 3\), a corollary of the completeness of the result.

Reviewer: N.Hitchin

### MSC:

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

32L25 | Twistor theory, double fibrations (complex-analytic aspects) |